Exact solutions can be achieved in the mathematical analysis of structures of permanent magnets under ideal conditions of linear demagnetization characteristics and for some special geometries and distributions of -magnetization. For instance, an exact mathematical procedure can be followed to design a magnet to generate a uniform field in an arbitrarily assigned polyhedral cavity with perfectly rigid magnetic materials and ideal ferromagnetic materials of infinite permeability.
In general, for arbitrary geometries and real characteristics of magnetic materials, only approximate numerical methods can be used to compute the field generated by a permanent magnet. The capability of handling systems of a large number of equations with modern computers has led to the development of powerful numerical tools such as the finite element methods, in which the domain of integration is divided in a large number of cells. By selecting a sufficiently small cell size, the variation of the field within each cell can be reduced to any desired level. Thus the integration of the Laplace's equation in each cell can be reduced to the dominant terms of a power series expansion and the constants of integration are determined by the boundary conditions at the interfaces between the cells. An iteration procedure is usually followed to solve the system of equations of the boundary conditions and the number of iterations depends on the required numerical precision of the result.
In applications where the field within the region of interest must be determined with extremely high precision, the large number of iterations may become a limiting factor in the use of these numerical methods. It is beyond the scope of this disclosure to provide a detailed explanation of past techniques for this purpose.
A special situation is encountered in magnetic structures that make use of the rare earth permanent magnets that exhibit quasi linear demagnetization characteristics with values of the magnetic susceptibility small compared to unity. A magnetic structure composed of these materials and ferromagnetic media of high magnetic permeability can be analyzed with a mathematical procedure based on a perturbation of the solution obtained in the limit of zero susceptibility and infinite permeability.
Structures composed of ideal materials of linear magnetic characteristics present a special situation where an exact solution is formulated by computing the field generated by volume and surface charges induced by the distribution of magnetization at the boundaries or interfaces between the different materials.